An improved integral model for plane and round turbulent buoyant jets

Abstract

The integral momentum and tracer equations for the mean motion with the turbulence contribution in momentum and tracer fluxes are integrated on the centreline of either plane or round buoyant jets, using suitable assumptions for the spreading coefficients and a closing function, and unified first- and second-order solutions are derived in the entire buoyancy range for mean axial velocities and mean concentrations. Comparisons to experimental data in the literature validate the model and show that second-order solutions deviate less than first-order solutions. Both types are used in conjunction with the integral continuity and kinetic energy equations for the mean motion to determine the variation of the local Richardson and Froude numbers, dispersion ratio, bulk dilution, dilution ratio, entrainment coefficient and mean velocity, kinetic energy flux and its gradient for the mean motion; and the variations of these quantities are evaluated using reported experimental or theoretical data. Finally, the variation of the product of kinetic energy flux and the local Richardson number is examined and a universal constant for both plane and round buoyant jets is revealed, leading to a unified definition of the local Richardson number, which is independent of the flow and mixing geometry and could be useful. Simple computational programming and good overall agreement make the proposed model a very promising tool for laboratory and field studies, outfall design and validation of numerical models.

Curvilinear Coordinate System for Mathematical Analysis of Inclined Buoyant Jets Using the Integral Method

“…In order to take into account the turbulent contribution to the mean flow and buoyancy fluxes, the Reynolds substitution is considered according to the form = + , where f represents any dependent variable of the buoyant jet with a mean value and turbulent fluctuation . In such a condition, after averaging the terms of the governing equations, the contribution of the last term of the right part of (13) becomes negligible compared to the mean turbulent contribution and may be omitted [5,9]. A common approximation for buoyant jet flows is the Boussinesq's approximation made for small initial density differences, i.e., when / 0 ≈ 1 and Δ 0 = − 0 << 0 ≤ .…”

“…Thus, the local fluid density is considered nearly constant and equal to a reference density r , where it is taken as either r = 0 or r = a . The usual assumptions and approximations are described in more detail elsewhere [17][18][19], while [5,9,20] have included the second-order terms of the turbulence contribution to momentum and buoyancy fluxes. Following Batchelor [14] and considering (3), (4), and (10), the term k ⋅ ∇k of (13) can be written with respect to the local coordinates as…”

“…The integration of the terms of PDEs is managed using the Leibnitz rule for the terms regarding gradient to the main flow direction and the Green's theorem for the terms regarding the transverse gradients [15]. The contribution of the fluxes of the turbulent terms of momentum and tracer with respect to the main flow direction to the mean flow fluxes is empirically introduced by coefficients based on available experimental findings [5,9]. The cross-sectional area A occupied by the two-dimensional buoyant jet, where integration takes place, is spanned from = -B w up to = B w .…”

“…Alternatively, depending on the selection of the solution procedure and in case that the specific particles are in curvilinear motion, the above equations may be preferable to be expressed at a curvilinear coordinate system [3]. Although, several buoyant jet phenomena can be analyzed by the governing equations written in orthogonal Cartesian or cylindrical coordinate systems [4][5][6][7][8], there exist some specific phenomena that the use of a curvilinear orthogonal coordinate system is mandatory [9]. Such phenomena include buoyant jet flows created by inclined discharges, in which the transverse profiles of the mean axial velocity and mean concentration are approximated with Gaussian profiles.…”

Positively and Negatively Round Turbulent Buoyant Jets into Homogeneous Calm Ambient

“…There are only a few analytical solutions related mainly to vertical positively buoyant jets [1][2][3]. In numerical investigations the scientists use the entrainment equations of mass, momentum and buoyancy conservation which they integrate in one dimension, and compare the findings with earlier experiments.…”